Freshness inventory refers to a stocking system of products with a relatively short shelf life such that managing some measure of freshness is a central concern. Freshness inventory differs from perishable inventory in several ways. Perishable inventory has a binary (0-1) utility: zero utility after the expiration date and full utility before. The utility of freshness inventory, in contrast, dynamically decreases to zero over time.
The joint pricing and inventory replenishment model has been studied in the setting of durable goods inventory. The survey papers, Yano and Gilbert (Yano, C. A., S. M. Gilbert. 2003. Coordinated pricing and production/procurement decisions: a review. In: Managing Business Interfaces: Marketing, Engineering and Manufacturing Perspectives, A. Chakravarty and J. Eliashberg (eds.), Kluwer Academic Publishers), Elmaghraby and Keskinocak (Elmaghraby, W., P. Keskinocak. 2003. Dynamic pricing in the presence of inventory considerations: research overview, current practices and future directions. Management Science, 49 (10) 1287-1309), and Chan et al. (Chan, L. M. A., Z. J. Shen, D. Simchi-Levi, J. L. Swann. 2004 Coordination of pricing and inventory decisions: a survey and classification. In Supply Chain Analysis in the eBusiness Era, D. Simchi-Levi, D. Wu and Z. J. Shen (eds), Kluwer Academic Publishers) provide studies of the joint pricing and inventory replenishment model in the setting of durable goods inventory. Whitin (Whitin, T. M. 1955. Inventory control and price theory. Management Science, 2(1) 61-68), Porteus (Porteus, E. L. 1985a. Investing in reduced setups in the EOQ model. Management Science, 31(8) 998-1010), Rajan, Rakesh and Steinberg (Rajan, A., Rakesh, R. Steinberg. 1992. Dynamic pricing and ordering decisions by a monopolist. Management Science, 38(2) 240-262), among others, study demands that are deterministic functions of price. Whitin (Whitin, T. M. 1955. Inventory control and price theory. Management Science, 2(1) 61-68) connects pricing and inventory control in the EOQ framework, and Porteus (Porteus, E. L. 1985a. Investing in reduced setups in the EOQ model. Management Science, 31(8) 998-1010) provides an explicit solution for the linear demand instance. Rajan, Rakesh and Steinberg (Rajan, A., Rakesh, R. Steinberg. 1992. Dynamic pricing and ordering decisions by a monopolist. Management Science, 38(2) 240-262) investigate continuous pricing for perishable products for which demands may diminish as products age.
The body of research on pricing and stochastic inventory control has been focused on establishing the optimality of structural inventory and pricing policies. Federgruen and Heching (Federgruen, A., A. Heching. 1999. Combined pricing and inventory control under uncertainty. Operations Research, 47(3) 454-475) examine a periodic-review model in which the demand in each period depends on the price charged in that period and a random term. The dependence can be quite general, but every realization of demand function is assumed to be concave in price. The replenishment cost is linear, without a fixed setup cost. The authors show that a base-stock list-price policy is optimal for both average and discounted objectives. Earlier related works include those by Zabel (Zabel, E. 1972. Multiperiod monopoly under uncertainty. Journal of Economic Theory, 5(3) 524-536) and Thowsen (Thowsen, G. T. 1975. A dynamic, nonstationary inventory problem for a price/quantity setting firm. Navel Research Logistics Quarterly, 22 461-476).
When including a replenishment setup cost, the relevant policy is the (s, S, p) policy (i.e., inventory control follows the usual (s, S) policy, while the price p depends on the initial inventory level. The optimality of (s, S, p) policy has been established under various settings. Periodic-review backorder setting is considered in Chen and Simchi-Levi (Chen, X., D. Simchi-Levi. 2004a. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6) 887-896 and Chen, X., D. Simchi-Levi. 2004b. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the infinite horizon case. Mathematics of Operations Research, 29(3) 698-723), Feng and Chen (Feng, Y., F. Y. Chen. 2004. Optimality and optimization of a joint pricing and inventory-control policy for a periodic-review system. Working paper, Chinese University of Hong Kong). Periodic-review lost sales setting is considered in Polatoglu and Sahin (Polatoglu, H., I. Sahin. 2000. Optimal procurement policies under price-dependent demand. International Journal of Production Economics, 65(2) 141-171), Chen, Ray and Song (Chen, F. Y., S. Ray, Y. Song. 2006. Optimal pricing and inventory control policy in periodic review systems with fixed ordering cost and lost sales. Naval Research Logistics, 53(2) 117-136). Huh and Janakiraman (Huh, W. T., G. Janakiraman. 2005. Optimality results in inventory-pricing control: an alternate approach. Working paper, Columbia University, New York University) provide an approach for proving and generalizing many of the early results for both backorder and lost sales settings. Continuous-review models are studied by Feng and Chen (Feng, Y., F. Y. Chen. 2003. Joint pricing and inventory control with setup costs and demand uncertainty. Working paper, Chinese University of Hong Kong), and Chen and Simchi-Levi (Chen, X., D. Simchi-Levi. 2006. Coordinating inventory control and pricing strategies: the continuous review model. Operations Research Letters, 34(3) 323-332).
One of the key determinants of the complexity and generality of these models is the assumption about the demand. The demand function in each period typically consists of a deterministic demand function and a random component. These two components can be additive (e.g., Chen, X., D. Simchi-Levi. 2004a. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6) 887-896 and Chen, F. Y., S. Ray, Y. Song. 2006. Optimal pricing and inventory control policy in periodic review systems with fixed ordering cost and lost sales. Naval Research Logistics, 53(2) 117-136), or multiplicative (e.g., Song, Y., S. Ray, T. Boyaci. 2006. Optimal dynamic joint pricing and inventory control for multiplicative demand with fixed order costs. Working paper, McGill University), or both (e.g., Zabel, E. 1972. Multiperiod monopoly under uncertainty. Journal of Economic Theory, 5(3) 524-536, Chen, X., D. Simchi-Levi. 2004a. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case. Operations Research, 52(6) 887-896 and Chen, X., D. Simchi-Levi. 2004b. Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the infinite horizon case. Mathematics of Operations Research, 29(3) 698-723). Some other models allow the random component to affect the demand in more general forms (e.g., Federgruen, A., A. Heching. 1999. Combined pricing and inventory control under uncertainty. Operations Research, 47(3) 454-475, Polatoglu, H., I. Sahin. 2000. Optimal procurement policies under price-dependent demand. International Journal of Production Economics, 65(2) 141-171, Feng, Y., F. Y. Chen. 2004. Optimality and optimization of a joint pricing and inventory-control policy for a periodic-review system. Working paper, Chinese University of Hong Kong, and Huh, W. T., G. Janakiraman. 2005. Optimality results in inventory-pricing control: an alternate approach. Working paper, Columbia University, New York University). In Chen, Wu and Yao (Chen, H., Wu, O. and Yao, D. D. 2010. On the Benefit of Inventory-Based Dynamic Pricing Strategies. Production and Operations Management, 19, 249-260), the demand model is Brownian motion with the drift and diffusion coefficients both depending on the price set by the firm. This price dependence is general enough to include the continuous-time analogy of additive demand and multiplicative demand as special cases. Furthermore, the diffusion coefficient brings out explicitly the impact of demand variability on various performance measures.
Another issue of interest is to quantify the profit improvement of using dynamic pricing over static pricing. Results along this line were mostly limited to numerical studies. Federgruen and Heching (Federgruen, A., A. Heching. 1999. Combined pricing and inventory control under uncertainty. Operations Research, 47(3) 454-475) experimented with a multiplicative demand case in a periodic-review system, and found that the benefit of dynamic pricing increases as demand variability increases. They reported a maximum of 6.54% increase in profit compared to a fixed pricing strategy. With a order-setup cost, Feng and Chen (Feng, Y., F. Y. Chen. 2004. Optimality and optimization of a joint pricing and inventory-controlpolicy for a periodic-review system. Working paper, Chinese University of Hong Kong) show that the profit improvement of dynamic pricing is limited (the profit improvement as a percentage of static pricing profit could be large when the static pricing profit is low.) For a periodic-review system with additive demand and lost sales, Chen, Ray and Song (Chen, F. Y., S. Ray, Y. Song. 2006. Optimal pricing and inventory control policy in periodic review systems with fixed ordering cost and lost sales. Naval Research Logistics, 53(2) 117-136) find that the profit improvement of dynamic pricing increases in the fixed ordering cost. They reported a maximum of 3.74% improvement in profit. Most of the above results exhibit rather modest benefits of dynamic pricing with respect to static pricing (when the demand is stationary). Gayon and Dallery (Gayon J. P., Y. Dallery. 2006. Dynamic vs static pricing in a make-to-stock queue with partially controlled production. OR Spectrum, forthcoming) point out that dynamic pricing is potentially much more beneficial when the replenishment process is partially controlled. Chen, Wu and Yao (Chen, H., Wu, O. and Yao, D. D. 2010. On the Benefit of Inventory-Based Dynamic Pricing Strategies. Production and Operations Management, 19, 249-260) develop an upper-bound to quantify the profit improvement using dynamic pricing and to identify situations where the improvement can be significant.